[Greg Boyles]

Can some one please briefly explain why natural cosmological events are unlikely to generate prime numbers.

[DrRocket]

Greg Boyles, on 5 December 2011 - 11:07 PM, said:

Can some one please briefly explain why natural cosmological events are unlikely to generate prime numbers.

What in the world is that supposed to mean ?

How would a "cosmological event" generate any number ? Physical events generate numbers with attendant units and you can make the bate number anything that you want by simply choosing the numbers appropriately. The exception are a very few fundamental constants (like the fine structue consytant) that occur as pure ratios and they are generally not integers at all.

This post has been edited by DrRocket: 5 December 2011 - 11:19 PM

[Greg Boyles]

DrRocket, on 5 December 2011 - 11:15 PM, said:

What in the world is that supposed to mean ?

How would a "cosmological event" generate any number ? Physical events generate numbers with attendant units and you can make the bate number anything that you want by simply choosing the numbers appropriately. The exception are a very few fundamental constants (like the fine structue consytant) that occur as pure ratios and they are generally not integers at all.

I am refering to the novel 'Contact' by Carl Sagan, in which he writes that prime numbers are unlikely to be generated by natral events.

Quote

How would a "cosmological event" generate any number ?

I have absolutely no idea. Not being a trained mathematician or physicist, I have no context in which to place this concept in order to judge its validity.

This post has been edited by Greg Boyles: 5 December 2011 - 11:32 PM

[IM Egdall]

I think what Sagan was getting at is this.

If you detected some kind of signal from outer space and say its strength varied over time -- and if you measured this strength you would get a series of values. If you looked at these values and they were proportional to the prime numbers, then it would most likely be an indication of some kind of message from an intelligent being, rather then the results of a natural phenomenon. (Prime numbers are those which cannot be factored into two or more numbers -- like 1, 2, 3, 5, 7, 11, 13, 17, 19 etc.)

This post has been edited by IM Egdall: 5 December 2011 - 11:48 PM

[Greg Boyles]

IM Egdall, on 5 December 2011 - 11:47 PM, said:

I think what Sagan was getting at is this.

If you detected some kind of signal from outer space and say its strength varied over time -- and if you measured this strength you would get a series of values. If you looked at these values and they were proportional to the prime numbers, then it would most likely be an indication of some kind of message from an intelligent being, rather then the results of a natural phenomenon. (Prime numbers are those which cannot be factored into two or more numbers -- like 1, 2, 3, 5, 7, 11, 13, 17, 19 etc.)

But why prime numbers? Why wouldn't a sequence of 1, 2, 4, 8, 16, 32,..... or 1, 1, 2, 3, 5, 8, 13,... be an equal indication of intelligence?

Would it not be the fact that the sequence of numbers is mathematically meaningful in some way regardless of whether it consisted of prime numbers or not?

This post has been edited by Greg Boyles: 6 December 2011 - 10:05 AM

[imatfaal]

A sequence 1,2,4,8,16,32 could easily be explained by natural means; a doubling of emitters or halving of interference would give a similar series. 1,1,2,3,5 could be generated by a signaller that both produces more signallers and decays after givingout two signals. we do not have a method of generating the series of primes

[Greg Boyles]

imatfaal, on 6 December 2011 - 11:53 AM, said:

A sequence 1,2,4,8,16,32 could easily be explained by natural means; a doubling of emitters or halving of interference would give a similar series. 1,1,2,3,5 could be generated by a signaller that both produces more signallers and decays after givingout two signals.

Can you cite any terrestrial or cosmic examples of these?

Particularly with the Fibonacci sequence, I have great difficulty in conceiving of a natural example of this that does not some how involve life.

[Dekan]

Erastosthenes' Sieve will generate the series of primes.

I used to write computer programs in BASIC, back in the 1980s, employing the Sieve to generate prime numbers ad infinitum. And trying to find a pattern in the numbers - which could produce a formula, to predict whether any random number, was Prime or not.

I never found the formula. And no human has managed to find it - yet.

But aliens might have done it! If so, mightn't they regard our transmitting Prime Numbers, as evidence of our dumbness. And hence, our unworthiness for admission to the Galactic Confederation?

This post has been edited by Dekan: 6 December 2011 - 12:31 PM

[Schrödinger's hat]

Greg Boyles, on 6 December 2011 - 12:20 PM, said:

Can you cite any terrestrial or cosmic examples of these?

Particularly with the Fibonacci sequence, I have great difficulty in conceiving of a natural example of this that does not some how involve life.

The fibonacci sequence comes up a lot in nature. Or at least sequences (or continuous things) that are very similar mathematically.
Look up logarithmic spirals. Objects that are self-similar will generate these and it wouldn't take much of a coincidence for the growth rate to approximate the fibonacci sequence.

Anything that is changes in a way proportional to itself will have a doubling time, and if that doubling time happened to match up with some other timing, you could easily get 2,4,8... or similar.

Many patterns that come from a simple recurrence relation or differential equation will tend to pop up in nature.


Prime numbers on the other hand are notoriously hard to weed out.
Other than some advanced methods, some of which which aren't deterministic, the only way to find prime numbers is to start with some prime numbers, then use those to figure out which numbers aren't prime and cross them off to find more (and thus repeat the process).
They don't arise easily from feedback loops and the nice (and sometimes not quite so nice, but still fairly tidy in the scheme of things) types of equation that natural phenomena tend to follow.

[Greg Boyles]

Schrödinger, on 6 December 2011 - 12:35 PM, said:

The fibonacci sequence comes up a lot in nature. Or at least sequences (or continuous things) that are very similar mathematically.
Look up logarithmic spirals. Objects that are self-similar will generate these and it wouldn't take much of a coincidence for the growth rate to approximate the fibonacci sequence.

Anything that is changes in a way proportional to itself will have a doubling time, and if that doubling time happened to match up with some other timing, you could easily get 2,4,8... or similar.

Many patterns that come from a simple recurrence relation or differential equation will tend to pop up in nature.


Prime numbers on the other hand are notoriously hard to weed out.
Other than some advanced methods, some of which which aren't deterministic, the only way to find prime numbers is to start with some prime numbers, then use those to figure out which numbers aren't prime and cross them off to find more (and thus repeat the process).
They don't arise easily from feedback loops and the nice (and sometimes not quite so nice, but still fairly tidy in the scheme of things) types of equation that natural phenomena tend to follow.

Fibonacci is associated intimately with life but not necessarily intelligence.

But we are really talking about radio pulses being received from interstellar space.

What natural phenomena would generate radio pulses, or pulses of some other form of energy, that form the sequence 2, 4, 8,...... or 1, 1, 2, 3, 5, 8,.......

[imatfaal]

Greg Boyles, on 6 December 2011 - 12:55 PM, said:

Fibonacci is associated intimately with life but not necessarily intelligence.

But we are really talking about radio pulses being received from interstellar space.

What natural phenomena would generate radio pulses, or pulses of some other form of energy, that form the sequence 2, 4, 8,...... or 1, 1, 2, 3, 5, 8,.......

Greg
I think the idea is that theoretically we can come up with explanations for doubling sequences ( a pulsar behind a cloud that loses half its density every period) and for most other sequences; for a series of prime numbers we do not know of a way to generate them in sequence (the only way is to check each number and move on).

This is to say that mathematically we can find ways to produce the sequence 1,2,4,8,16... and from that mathematical base we can envisage physical methods (unlikely perhaps but still realistic) - however we do no know of a mathematical base which allows us to move from one prime to the next prime (or to generate the nth prime) - and thus we cannot see anyway that a physical process would mimic that series, therefore some intelligence must be behind it

[Greg Boyles]

imatfaal, on 6 December 2011 - 03:41 PM, said:

Greg
I think the idea is that theoretically we can come up with explanations for doubling sequences ( a pulsar behind a cloud that loses half its density every period) and for most other sequences; for a series of prime numbers we do not know of a way to generate them in sequence (the only way is to check each number and move on).

This is to say that mathematically we can find ways to produce the sequence 1,2,4,8,16... and from that mathematical base we can envisage physical methods (unlikely perhaps but still realistic) - however we do no know of a mathematical base which allows us to move from one prime to the next prime (or to generate the nth prime) - and thus we cannot see anyway that a physical process would mimic that series, therefore some intelligence must be behind it

OK, so all this pretty much boils down to the fact that Carl Sagan has assumed that extraterrestrial cultures have not discovered a mathematical means of generating prime numbers and that they would place the same signficance on them as our mathematicians do.

I am not discounting it, but it places the prime number thing into a context that I can comprehend a little better.

The point being that, if radio astronomers did discover a fibonacci or doubling sequence being pulsed out from a distant star system, then they would not necessarily dismiss it out of hand as being of natural origin simply because there is a mathematical means of generating it.

This post has been edited by Greg Boyles: 6 December 2011 - 11:01 PM

[imatfaal]

Greg Boyles, on 6 December 2011 - 10:55 PM, said:

OK, so all this pretty much boils down to the fact that Carl Sagan has assumed that extraterrestrial cultures have not discovered a mathematical means of generating prime numbers and that they would place the same signficance on them as our mathematicians do.

I am not discounting it, but it places the prime number thing into a context that I can comprehend a little better.

The point being that, if radio astronomers did discover a fibonacci or doubling sequence being pulsed out from a distant star system, then they would not necessarily dismiss it out of hand as being of natural origin simply because there is a mathematical means of generating it.

Greg - you are being a bit too dogmatic about it. Any unexplained signal will be investigated - and few people who think of themselves as scientists dismiss ANYTHING out of hand. When the signals from pulsar were first discovered they were labelled LGM1, LGM2.... the LGM stood half jokingly for little green man.

Irregular non-repeating noise comes at us from all angles - from within that noise we seek any semblance of order or of progression as this may be a sign of intelligence. Some non-random sequences, progressions etc are easier to explain from astronomical reasons than others. We cannot even conceive of a method of naturally generating the primes (with each term being generated by the previous terms) thus Sagan correctly says if we saw that sequence we would have to believe it was non-natural.

[Schrödinger's hat]

imatfaal, on 7 December 2011 - 11:39 AM, said:

Sagan correctly says if we saw that sequence we would have to believe it was non-natural.

Would have to is possibly slightly strong, but it would be a very clear signal that something interesting was happening.
Exponentials (I include both and the fibonacci sequence in this, as it can be described as the sum of two numbers, each raised to the same exponent) on the other hand pop up all over the place.
Even if some alien race had discovered natural phenomena that generated prime numbers they would probably still illicit interest.
There are many other sequences we could use (ratios of masses of the most fundamental particles we know of for example, or something involving common fractals, or some encoding of pi¹), but it's a fairly good bet anyone advanced enough to have a radio telescope (or any hobbiest looking through one in a civilisation where they are more widely available) would recognise prime numbers, and not relate them to common natural phenomena.
Anything we pick would potentially look terribly naive to some highly advanced race, including the use of radio waves. If they were interested in responding I doubt our primitive mathematics would be any further deterrant.


¹These would require some higher level encoding. Integers such as primes could be a series of beeps, or some other discrete part of the signal. Decimals or irrationals require further context. You can't just assume they use base ten. Any race as or more advanced would not have too much trouble recognising these numbers in any base if they were looking, but they would not stand out from the noise nearly as well as a sequence of integers.
Other advantages of primes are they are monotonically increasing, increase quite rapidly in size (exponentials share this one), and once you get across the idea of primes you can use composite numbers to try and get across the idea that you're sending an image.

[imatfaal]

Schrödinger, on 7 December 2011 - 11:59 AM, said:

Would have to is possibly slightly strong, but it would be a very clear signal that something interesting was happening.

I thought hard before I wrote that - so let me put down a bit of my reasoning; we have been searching for over two thousand years for a form of progression that will generate primes and we have not found it. that a natural phenomenon would generate a sequence that we cannot derive mathematically seems so unlikely that we should (in the absence of contrary evidence or contrary advances in maths) conclude that the generator of such a sequence is non-natural

Quote

Exponentials (I include both and the fibonacci sequence in this, as it can be described as the sum of two numbers, each raised to the same exponent) on the other hand pop up all over the place.
Even if some alien race had discovered natural phenomena that generated prime numbers they would probably still illicit interest.
There are many other sequences we could use (ratios of masses of the most fundamental particles we know of for example, or something involving common fractals, or some encoding of pi¹), but it's a fairly good bet anyone advanced enough to have a radio telescope (or any hobbiest looking through one in a civilisation where they are more widely available) would recognise prime numbers, and not relate them to common natural phenomena.
Anything we pick would potentially look terribly naive to some highly advanced race, including the use of radio waves. If they were interested in responding I doubt our primitive mathematics would be any further deterrant.


¹These would require some higher level encoding. Integers such as primes could be a series of beeps, or some other discrete part of the signal. Decimals or irrationals require further context. You can't just assume they use base ten. Any race as or more advanced would not have too much trouble recognising these numbers in any base if they were looking, but they would not stand out from the noise nearly as well as a sequence of integers.
Other advantages of primes are they are monotonically increasing, increase quite rapidly in size (exponentials share this one), and once you get across the idea of primes you can use composite numbers to try and get across the idea that you're sending an image.

On irrationals I believe you are incorrect - but I recognise your maths is way advanced of mine; so let me run this one past. theoretically, surely there are arbitrarily long sections of pi in one number base that are repeated in e in another number base or in the same base; due to the fact that they are non-repeating and infinite. in fact any selection of numbers of any length will be found in any irrational if one extends that irrational long enough. the point being that you have to be there observing at the beginning of the signal and that the irrationals are only transimitted in a truncated format. come to think of it that is the only way that any intelligent race would transmit - prolly best to ignore above para

[Greg Boyles]

Quote

Exponentials (I include both and the fibonacci sequence in this, as it can be described as the sum of two numbers, each raised to the same exponent) on the other hand pop up all over the place.

Can anyone provide a physics example, involving pulses of some sort of energy, of such sequences popping up?

There are a lot of things in science that I don't fully understand and take the word of the experts in the relevant fields.

But the film 'Contact' made a huge impression on me and now that I am reading the book, I would like to comprehend more fully the cosmic signficance of prime numbers.

I now understand the mathematics of them a little better, but to round off my understanding, I would like to understand how other number sequences I have mentioned can be generated by naturally. A concrete physical example rather than mathematical theory.

This post has been edited by Greg Boyles: 7 December 2011 - 01:13 PM

[Schrödinger's hat]

imatfaal, on 7 December 2011 - 12:14 PM, said:

I thought hard before I wrote that - so let me put down a bit of my reasoning; we have been searching for over two thousand years for a form of progression that will generate primes and we have not found it. that a natural phenomenon would generate a sequence that we cannot derive mathematically seems so unlikely that we should (in the absence of contrary evidence or contrary advances in maths) conclude that the generator of such a sequence is non-natural

I was merely presenting a weaker form of the same argument.
Regardless of whether or not a natural generator were found (I agree that this is highly unlikely given current knowledge), the fact we have not done so would make one interesting even if one did not assume there were intelligence behind it initially. As such, further investigation would be likely and any intelligence behind it would become more apparent.

Quote

On irrationals I believe you are incorrect - but I recognise your maths is way advanced of mine; so let me run this one past. theoretically, surely there are arbitrarily long sections of pi in one number base that are repeated in e in another number base or in the same base; due to the fact that they are non-repeating and infinite. in fact any selection of numbers of any length will be found in any irrational if one extends that irrational long enough. the point being that you have to be there observing at the beginning of the signal and that the irrationals are only transimitted in a truncated format. come to think of it that is the only way that any intelligent race would transmit - prolly best to ignore above para

Well you said ignore it, but I didn't notice until I had already written a response, so here it is anyway as I thought it might be interesting.
Obviously such numbers would have to be truncated at some point.
A good way around the beginning thing is to repeat it.
If I had reason to believe someone was trying to communicate with me, and I identified some kind of restricted character set, one thing I would try is a variety of number bases.
Here's one for you to try
dfhfaachdcgehhcdeahbdehgafhdf
Granted you already know the general scheme, but it's not too much of a head start for a proof of concept.

This would probably be a very poor scheme, especially with longer sections of transcendental numbers, as they could appear very much like noise.

Alternatively I could encode something like pi as a series of rational approximations

... -
............. ----
................ -----
................... ------
...................... -------

................................................................................................................................................................................... ---------------------------------------------------------

and so on.
This is much more clearly a message, and would probably be about as effective as the prime numbers at getting the point across. Although it lacks the advantage of perparing the reader for the use of composite numbers to delineate area-like and volume-like entities (such as images).

[Airbrush]

A series of prime numbers is not the only way an ETI could signal their existence. (BTW, why would they? It's like saying "Here we are, come and bother us.")

Many mathematical number sequences could do the same job, such as some kind of Morris Code.

This post has been edited by Airbrush: 9 December 2011 - 12:14 AM

[imatfaal]

Airbrush, on 8 December 2011 - 08:01 PM, said:

A series of prime numbers is not the only way an ETI could signal their existence. (BTW, why would they? It's like saying "Here we are, come and bother us.")

Many mathematical number sequences could do the same job, such as some kind of Morris Code.

Morris Code - nope lost me there (morse?) - the whole point of using primes is that we cannot at present see it as anything but artificial and with an intelligence behind it - most mathematical sequences could be natural

[Airbrush]

imatfaal, on 9 December 2011 - 06:40 PM, said:

Morris Code - nope lost me there (morse?) - the whole point of using primes is that we cannot at present see it as anything but artificial and with an intelligence behind it - most mathematical sequences could be natural

Thanks for the spelling. "Most math sequences could be natural." But not all. Rather, few are natural. In fact, there should be an infinite variety of sequences that are obviously not natural. A sequence of prime numbers is just one of them. All you need is a pattern that is obviously not natural. Dots and dashes that are in a complicated, but ordered sequence, is a start.

How about a complicated mathematical equation that repeats, or it changes slightly in such a way that the changes are in an artificial pattern. Or the structure of the DNA molecule that repeats? The list should go on and on.

This post has been edited by Airbrush: 9 December 2011 - 08:08 PM